Solve − 9.4 ≥ 1.7 X + 4.2 -9.4 \geq 1.7x + 4.2 − 9.4 ≥ 1.7 X + 4.2 A. X ≤ − 8 X \leq -8 X ≤ − 8 B. X ≤ 8 X \leq 8 X ≤ 8 C. X ≥ − 8 X \geq -8 X ≥ − 8 D. X ≥ 8 X \geq 8 X ≥ 8

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Introduction

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Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving linear inequalities of the form $ax + b \geq cx + d$, where $a$, $b$, $c$, and $d$ are constants. We will use the given problem, $-9.4 \geq 1.7x + 4.2$, as a case study to illustrate the steps involved in solving linear inequalities.

Understanding the Problem

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The given problem is a linear inequality of the form $-9.4 \geq 1.7x + 4.2$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The goal is to find the values of $x$ that satisfy the inequality.

Step 1: Subtract 4.2 from Both Sides

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The first step in solving the inequality is to subtract 4.2 from both sides of the inequality. This will help us isolate the term involving $x$.

-9.4 \geq 1.7x + 4.2
-9.4 - 4.2 \geq 1.7x + 4.2 - 4.2
-13.6 \geq 1.7x

Step 2: Divide Both Sides by 1.7

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Next, we need to divide both sides of the inequality by 1.7 to solve for $x$. This will give us the values of $x$ that satisfy the inequality.

-13.6 \geq 1.7x
\frac{-13.6}{1.7} \geq \frac{1.7x}{1.7}
-8 \geq x

Analyzing the Solution

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Now that we have solved the inequality, let's analyze the solution. The inequality $-8 \geq x$ tells us that $x$ is less than or equal to -8. This means that any value of $x$ that is less than or equal to -8 will satisfy the inequality.

Conclusion

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In conclusion, solving linear inequalities involves a series of steps, including subtracting constants from both sides and dividing both sides by coefficients. By following these steps, we can isolate the variable and find the values that satisfy the inequality. In this article, we used the problem $-9.4 \geq 1.7x + 4.2$ as a case study to illustrate the steps involved in solving linear inequalities.

Frequently Asked Questions

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Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression of the form $ax + b$, where $a$ and $b$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves subtracting constants from both sides and dividing both sides by coefficients.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, whereas a linear inequality is an inequality that involves a linear expression.

Examples

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Example 1: Solving the Inequality $2x + 3 \geq 5$

To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

2x + 3 \geq 5
2x \geq 5 - 3
2x \geq 2
x \geq \frac{2}{2}
x \geq 1

Example 2: Solving the Inequality $x - 2 \leq 3$

To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

x - 2 \leq 3
x \leq 3 + 2
x \leq 5

Tips and Tricks

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Tip 1: Always Isolate the Variable

When solving a linear inequality, it's essential to isolate the variable on one side of the inequality sign. This will help you find the values that satisfy the inequality.

Tip 2: Use Inverse Operations

To solve a linear inequality, you need to use inverse operations to isolate the variable. For example, if you have an inequality of the form $ax + b \geq cx + d$, you can subtract $b$ from both sides and then divide both sides by $a$ to solve for $x$.

Tip 3: Check Your Solution

Once you've solved the inequality, it's essential to check your solution to ensure that it satisfies the original inequality.

Final Thoughts

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Solving linear inequalities is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve linear inequalities with confidence. Remember to always isolate the variable, use inverse operations, and check your solution to ensure that it satisfies the original inequality. With practice and patience, you'll become proficient in solving linear inequalities in no time.

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Introduction

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Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will address some of the most frequently asked questions about linear inequalities, providing clear and concise answers to help you better understand this topic.

Q&A

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Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression of the form $ax + b$, where $a$ and $b$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves subtracting constants from both sides and dividing both sides by coefficients.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, whereas a linear inequality is an inequality that involves a linear expression.

Q: Can I use the same methods to solve linear inequalities as I do to solve linear equations?

A: While some methods are similar, you cannot use the same methods to solve linear inequalities as you do to solve linear equations. Linear inequalities require a different approach, as you need to consider the direction of the inequality sign.

Q: How do I determine the direction of the inequality sign?

A: The direction of the inequality sign depends on the coefficient of the variable. If the coefficient is positive, the inequality sign is "greater than or equal to" (≥). If the coefficient is negative, the inequality sign is "less than or equal to" (≤).

Q: Can I use inverse operations to solve linear inequalities?

A: Yes, you can use inverse operations to solve linear inequalities. For example, if you have an inequality of the form $ax + b \geq cx + d$, you can subtract $b$ from both sides and then divide both sides by $a$ to solve for $x$.

Q: How do I check my solution to a linear inequality?

A: To check your solution, substitute the value of $x$ back into the original inequality and verify that it satisfies the inequality.

Q: Can I use a calculator to solve linear inequalities?

A: While a calculator can be a useful tool, it's not always necessary to solve linear inequalities. In many cases, you can solve the inequality by hand using basic algebraic operations.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

• Not isolating the variable on one side of the inequality sign
• Not considering the direction of the inequality sign
• Not using inverse operations correctly
• Not checking the solution

Examples

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Example 1: Solving the Inequality $2x + 3 \geq 5$

To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

2x + 3 \geq 5
2x \geq 5 - 3
2x \geq 2
x \geq \frac{2}{2}
x \geq 1

Example 2: Solving the Inequality $x - 2 \leq 3$

To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

x - 2 \leq 3
x \leq 3 + 2
x \leq 5

Tips and Tricks

---

Tip 1: Always Isolate the Variable

When solving a linear inequality, it's essential to isolate the variable on one side of the inequality sign. This will help you find the values that satisfy the inequality.

Tip 2: Use Inverse Operations

To solve a linear inequality, you need to use inverse operations to isolate the variable. For example, if you have an inequality of the form $ax + b \geq cx + d$, you can subtract $b$ from both sides and then divide both sides by $a$ to solve for $x$.

Tip 3: Check Your Solution

Once you've solved the inequality, it's essential to check your solution to ensure that it satisfies the original inequality.

Final Thoughts

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Solving linear inequalities requires a clear understanding of the concept and the ability to apply the necessary algebraic operations. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear inequalities. Remember to always isolate the variable, use inverse operations, and check your solution to ensure that it satisfies the original inequality. With practice and patience, you'll become a master of solving linear inequalities in no time.